Does anyone know of a (small) company flying Guatemala City - Belize City or other destination in Belize regularly?

As far as a I know, there is none. Tropic Air goes twice daily between BZE and Flores. TACA and maybe still TAG fly Flores-Guatemala City.

TACA goes GUA to BZE via San Salvador, El Salvador, with a change of planes at TACA’s hub there. You could also fly GUA to BZE via Houston, also with a change of planes and a layover.

The most direct, and usually fastest and least expensive way is from GUA to Flores, and then from FRS to BZE. The cheapest way is via ADN bus from Guatemala City to Flores/Santa Elena, then Linea Dorada to Belize City.